nLab
Heisenberg n-group

Contents

Context

Symplectic geometry

Geometric quantization

Contents

Idea

A Heisenberg nn-group is a sub-∞-group of a quantomorphism n-group of a n-plectic space (G,ω)(G, \omega) equipped with ∞-group structure, on those whose underlying n-plectomorphisms act by lect ∞-action of GG on itself. This is the higher refinement of the traditional notion of Heisenberg group.

Definition

For (G,ω)(G, \omega) an n-plectic geometry with higher geometric prequantization (G,)(G, \nabla) and for GG equipped with ∞-group structure, the corresponding Heisenberg \infty-group is the sub-∞-group of the quantomorphism n-group on those elements whose corresponding n-plectomorphism is given by such a left action.

The corresponding Lie n-algebra is the Heisenberg Lie n-algebra.

Properties

Relation to parameterized WZW models

Examples

String 2-group is Heisenberg 2-group of WZW gerbe

For GG a compact simply connected simple Lie group, there is the “WZW gerbe”, hence the circle 2-bundle with connection on GG whose curvature 3-form is the left invariant extension θ[θθ]\langle \theta \wedge [\theta \wedge \theta]\rangle of the canonical Lie algebra 3-cocycle to the group

WZW:GB 2. \mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2 \,.
Proposition

The string 2-group is the smooth 2-groupo of automorphism of WZW\mathcal{L}_{WZW} which cover the left action of GG on itself (hence the “Heisenberg 2-group” of WZW\mathcal{L}_{WZW} regarded as a prequantum 2-bundle)

Aut( WZW)String(G), \mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,,

This is due to (Fiorenza-Rogers-Schreiber 13, section 6.2.1).

higher and integrated Kostant-Souriau extensions:

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

References

Last revised on October 17, 2013 at 22:35:46. See the history of this page for a list of all contributions to it.